Saturday, August 30, 2008

Saturday August 30, 2008

Filed under: General — m759 @ 6:28 PM
Ready when
you are, C. B.

“Hurricane Gustav is bearing down on the Gulf Coast, a reminder of Hurricane Katrina in 2005 and the Bush administration’s poor response. The storm was clearly on McCain’s mind Saturday.

‘You know, it just wouldn’t be appropriate to have a festive occasion while a near tragedy or a terrible challenge is presented in the form of a natural disaster….’ McCain said in an interview taped for ‘Fox News Sunday.'” —AP Aug. 30

Photos from John McCain’s
birthday three years ago:

The Gulf Coast,
Aug. 29, 2005:

The image “http://www.log24.com/log/pix06A/060829-Katrina.gif” cannot be displayed, because it contains errors.

The same day:

The image “http://www.log24.com/log/pix06A/060829-McCain.jpg” cannot be displayed, because it contains errors.

President George W. Bush
joins John McCain
in a celebration of
 McCain’s 69th birthday

“… liberal filmmaker Michael Moore… said Friday that the timing of Hurricane Gustav is ‘proof that there is a God in heaven,’ since the storm approaching the Gulf Coast could disrupt next week’s Republican National Convention.” —Fox News

From JohnMcCain.com
on August 1, 2008:


“Behold his mighty hand!”

Saturday August 30, 2008

Filed under: General — m759 @ 9:00 AM
Poetry and Politics* —


"… maybe it was McCain's role as 'movie-teller' that he cherishes most– the man who used to narrate the plots of films to his fellow PoWs in the compound. 'I must have told a hundred movies,' says McCain. 'Of course I don't know a hundred movies– I made them up.'"

The Guardian, quoted here on McCain's birthday, August 29, 2006. (McCain's birthday nine years earlier was the date of Judgment Day in "Terminator 2.")

A story from McCain's
birthday this year:


"Hail Sarah!"

Sarah Connor, mother of the savior in 'Terminator 2'

"At the still point,
there the dance is."

Four Quartets

"… the Four Quartets themselves appear, in all their complexity, as the poetry of simple civic virtue– the poetry of a poet trying to read the writing of the law that has become all but illegible. This, you may say, has nothing to do with poetry. On the contrary, it is one of the few truly hopeful signs that this civic virtue could once more be realized poetically."

— Erich Heller, quoted here
on August 25, 2008
(Feast of St. Louis)

Related material:

St. Sarah's Day,
The Dance:

See also the remarks of St. Augustine and others on time (August 28 entry) and, from May 24,  a song hook thanks to Cyndi Lauper:

* Also known as smoke and mirrors.

Thursday, August 28, 2008

Thursday August 28, 2008

Filed under: General — m759 @ 12:00 PM
See also the Log24 entry
on May 20, 2005,
the day Paul Ricoeur died.

Thursday August 28, 2008

Filed under: General,Geometry — m759 @ 5:24 AM
for the writer
known as UD


"Have liberty not as
     the air within a grave
Or down a well. Breathe freedom,
     oh, my native,
In the space of horizons
     that neither love nor hate."

— Wallace Stevens,
   "Things of August"

Remarks on physics, with apparently unrelated cartoon, New Yorker, Oct. 2, 2006

A related visual  
association of ideas —

("The association is the idea"
— Ian Lee, The Third Word War)

From UD Jewelry:

For  fishing enthusiasts: hook pendant from UD Jewelry

by John Braheny

"Hook" is the term you'll hear most often in the business and craft of commercial songwriting. (Well, maybe not as much as "Sorry, we can't use your song," but it's possible that the more you hear about hooks now, the less you'll hear "we can't use it" later.)

The hook has been described as "the part(s) you remember after the song is over," "the part that reaches out and grabs you," "the part you can't stop singing (even when you hate it)" and "the catchy repeated chorus…."

See also UD's recent
A Must-Read and In My Day*
as well as the five
Log24 entries ending
Sept. 20, 2002.

More seriously:
The date of The New Yorker issue quoted above is also the anniversary of the birth of Wallace Stevens and the date of death of mathematician Paul R. Halmos.
Stevens's "space of horizons" may, if one likes, be interpreted as a reference to projective geometry. Despite the bleak physicist's view of mathematics quoted above, this discipline is– thanks to Blaise Pascal— not totally lacking in literary and spiritual associations.

* Hey Hey

Wednesday, August 27, 2008

Wednesday August 27, 2008

Filed under: General — m759 @ 6:23 AM
“One Shot”
Keynote Address,
 National Convention

Of the People,
by the People,
for the People

From the autobiography of Reba McEntire:

“…my major field of study was elementary education and my minor was music. I received my bachelor’s degree, but never taught school as my Mama and Grandma had done before me….”  —My Story, Bantam, 1994

From a notable production of  “Annie Get Your Gun” starring Reba McEntire:

“Doin’ what comes naturally….”
— Irving Berlin

From Zenna Henderson’s first story of the People:

“Suddenly I felt her, so plainly that I knew with a feeling of fear and pride that I was of my grandmother, that soon I would be bearing the burden and blessing of her Gift — the Gift that develops into free access to any mind, one of the People or an Outsider, willing or not. And besides the access, the ability to counsel and help, to straighten tangled minds and snarled emotions…. It was the first time I had ever sorted anybody.”

— “Ararat,” in The Magazine of Fantasy & Science Fiction, October 1952 (reprinted in Ingathering, NESFA Press, 1995)

“You know, I spent 20 years in business. If you ran a company whose only strategy was to tear down the competition, it wouldn’t last long. So why is this wisdom so hard to find in Washington?

I know we’re at the Democratic convention, but if an idea works, it really doesn’t matter if it has an ‘R’ or ‘D’ next to it. Because this election isn’t about liberal versus conservative. It’s not about left versus right. It’s about the future versus the past.

In this election, at this moment in our history, we know what the problems are. We know that at this critical juncture, we have only one shot to get it right….

Let me tell you about a place called Lebanon– Lebanon, Virginia.”

— Last night’s keynote address at the Democratic National Convention

Related material

Map of Lebanon VA in relation to Bluefield WV, Pikeville KY, and Asheville NC

“The lunatic,
  the lover,
  and the poet
  are of imagination
  all compact.”
  — Shakespeare

For further details,
see the sons and
daughters of
Bluefield, Pikeville,
and Asheville.

Monday, August 25, 2008

Monday August 25, 2008

Filed under: General — Tags: — m759 @ 3:23 AM
For the Feast of
St. Louis

The concluding paragraph of Erich Heller's 1953 essay, "The Hazard of Modern Poetry"–

"'The poetry does not matter.' These words from Mr. T.S. Eliot's Four Quartets acquire an all but revolutionary significance if we understand them not only in their particular context but also in the context of a period of poetry in which nothing mattered except poetry. Against this background the Four Quartets themselves appear, in all their complexity, as the poetry of simple civic virtue– the poetry of a poet trying to read the writing of the law that has become all but illegible. This, you may say, has nothing to do with poetry. On the contrary, it is one of the few truly hopeful signs that this civic virtue could once more be realized poetically. For in speaking to the hazard of modern poetry I did not wish to suggest that the end had come for singers and skylarks. There will always be skylarks; perhaps even a few nightingales. But poetry is not only the human equivalent of the song of singing birds. It is also Virgil, Dante, and Hölderlin. It is also, in its own terms, the definition of the state of man."

Sunday, August 24, 2008

Sunday August 24, 2008

Filed under: General — m759 @ 7:00 AM

Yesterday’s entry, Absurdities, quoted Erich Heller:

“All relevant objective truths are born and die as absurdities. They come into being as the monstrous claim of an inspired rebel and pass away with the eccentricity of a superstitious crank.”

The context for this remarkable saying is Heller’s essay “The Hazard of Modern Poetry.” (See p. 270 in the links below.)

Discussing “the century of Pascal and Hobbes,” he says (see the link to p. 269 below) that

“… as for spiritual cunning, it was in the conceits of metaphysical poetry, in the self-conscious ambiguities of poetical language (there are, we are told, as many types of it as deadly sins), and in the paradoxes of Pascal’s religious thought. For ambiguity and paradox are the manner of speaking when reality and symbol, man’s mind and his soul, are at cross-purposes.”

Heller’s description of “relevant objective truths” as “absurdities” seems to be an instance of such ambiguity and paradox. For further details, see

The Disinherited Mind: Essays in Modern German Literature and Thought (Harvest paperback, 1975)–

“The Hazard of Modern Poetry” (pp. 263-300), Section 1, pages

263, 264, 265, 266, 267, 268, 269, 270, 271, 272.

For material related to Pascal, see the five Log24 entries ending on D-Day, 2008.

For material related to Hobbes, see the five Log24 entries ending on St. Patrick’s Day, 2007.

Saturday, August 23, 2008

Saturday August 23, 2008

Filed under: General — m759 @ 5:01 AM

“The balance-beam of Fate was bent;  
The bounds of good and ill were rent;  
Strong Hades could not keep his own,  
But all slid to confusion.”

— “Uriel,” by  
Ralph Waldo Emerson:

Oxford Book of
English Verse
, 1919,

“All relevant objective truths are born and die as absurdities. They come into being as the monstrous claim of an inspired rebel and pass away with the eccentricity of a superstitious crank.”

— Erich Heller, The Disinherited Mind

NY lottery Aug. 22, 2008: mid-day 670, evening 666

Related material:

Yesterday’s entry
Angels in Arabia

Friday, August 22, 2008

Friday August 22, 2008

Filed under: General,Geometry — m759 @ 5:01 AM

Tentative movie title:

Kohs Block Design Test

The Kohs Block Design
Intelligence Test

Samuel Calmin Kohs, the designer (but not the originator) of the above intelligence test, would likely disapprove of the "Aryan Youth types" mentioned in passing by a film reviewer in today's New York Times. (See below.) The Aryan Youth would also likely disapprove of Dr. Kohs.

Related material from
Notes on Finite Geometry:

Kohs Block Design figure illustrating the four-color decomposition theorem

Other related material:

1.  Wechsler Cubes (intelligence testing cubes derived from the Kohs cubes shown above). See…

Harvard psychiatry and…
The Montessori Method;
The Crimson Passion;
The Lottery Covenant.

2.  Wechsler Cubes of a different sort (Log24, May 25, 2008)

3.  Manohla Dargis in today's New York Times:

"… 'Momma’s Man' is a touchingly true film, part weepie, part comedy, about the agonies of navigating that slippery slope called adulthood. It was written and directed by Azazel Jacobs, a native New Yorker who has set his modestly scaled movie with a heart the size of the Ritz in the same downtown warren where he was raised. Being a child of the avant-garde as well as an A student, he cast his parents, the filmmaker Ken Jacobs and the artist Flo Jacobs, as the puzzled progenitors of his centerpiece, a wayward son of bohemia….

In American movies, growing up tends to be a job for either Aryan Youth types or the oddballs and outsiders…."

4.  The bohemian who named his son Azazel:

"… I think that the deeper opportunity, the greater opportunity film can offer us is as an exercise of the mind. But an exercise, I hate to use the word, I won't say 'soul,' I won't say 'soul' and I won't say 'spirit,' but that it can really put our deepest psychological existence through stuff. It can be a powerful exercise. It can make us think, but I don't mean think about this and think about that. The very, very process of powerful thinking, in a way that it can afford, is I think very, very valuable. I basically think that the mind is not complete yet, that we are working on creating the mind. Okay. And that the higher function of art for me is its contribution to the making of mind."

Interview with Ken Jacobs, UC Berkeley, October 1999

5.  For Dargis's "Aryan Youth types"–

From a Manohla Dargis
New York Times film review
of April 4, 2007
   (Spy Wednesday) —

Scene from Paul Verhoeven's film 'Black Book'

See also, from August 1, 2008
(anniversary of Hitler's
opening the 1936 Olympics) —

For Sarah Silverman

and the 9/9/03 entry 

Olympic Style.

August 21-22, 2008:


Wednesday, August 20, 2008

Wednesday August 20, 2008

Filed under: General — m759 @ 11:29 PM
For Madeleine L’Engle,
wherever she may be

The entries of yesterday (updated today) and the day before suggest a flashback to the five “Dungeons & Dragons” entries ending on March 6, 2008.  For more about dungeons, see Jan. 7, 2007. For more about dragons, see Crystal and Dragon: The Cosmic Dance of Symmetry and Chaos in Nature, Art and Consciousness, by David Wade.

Tuesday, August 19, 2008

Tuesday August 19, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 8:30 AM
Three Times

"Credences of Summer," VII,

by Wallace Stevens, from
Transport to Summer (1947)

"Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed
The object, grips it
     in savage scrutiny,
Once to make captive,
     once to subjugate
Or yield to subjugation,
     once to proclaim
The meaning of the capture,
     this hard prize,
Fully made, fully apparent,
     fully found."

Stevens does not say what object he is discussing.

One possibility —

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:

"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."

Another possibility —

The 4x4 square

  A more modest object —
the 4×4 square.

Update of Aug. 20-21 —

Symmetries and Facets

Kostant's poetic comparison might be applied also to this object.

The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."

More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.

For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.

* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–

Platonic solids' symmetry groups

The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).

If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that

"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."

The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.

Monday, August 18, 2008

Monday August 18, 2008

Filed under: General,Geometry — m759 @ 9:00 AM
The Revelation Game

(See also Jung’s birthday.)

Google logo, Aug. 18, 2008: Dragon playing Olympic ping pong

Lotteries on
August 17,
(No revelation)
New York
(No belief)
No belief,
no revelation



4 9 2
3 5 7
8 1 6

(See below.)

without belief



in Toronto

Belief without


on Grace

(See below.)

Belief and



and Art

No belief, no revelation:
An encounter with “492”–

“What is combinatorial mathematics? Combinatorial mathematics, also referred to as combinatorial analysis or combinatorics, is a mathematical discipline that began in ancient times. According to legend the Chinese Emperor Yu (c. 2200 B.C.) observed the magic square

4 9 2
3 5 7
8 1 6

on the shell of a divine turtle….”

— H.J. Ryser, Combinatorial Mathematics, Mathematical Association of America, Carus Mathematical Monographs 14 (1963)

Belief without revelation:
Theology and human experience,
and the experience of “272”–

From Christian Tradition Today,
by Jeffrey C. K. Goh
(Peeters Publishers, 2004), p. 438:

“Insisting that theological statements are not simply deduced from human experience, Rahner nevertheless stresses the experience of grace as the ‘real, fundamental reality of Christianity itself.’ 272

272  ‘Grace’ is a key category in Rahner’s theology.  He has expended a great deal of energy on this topic, earning himself the title, amongst others, of a ‘theologian of the graced search for meaning.’ See G. B. Kelly (ed.), Karl Rahner, in The Making of Modern Theology series (Edinburgh: T&T Clark, 1992).”

Sunday, August 17, 2008

Sunday August 17, 2008

Filed under: General — m759 @ 6:20 AM
Maybe Escher
could have done it.

Escher, 'Verbum,' detail

Detail from

(“In Touch with God“)

The title link of this entry
leads, via a Log24 entry, to
a story by Robert A. Heinlein.

For those who, like Rick Warren
(shown below in a current news page),

TIME photo of preacher Rick Warren embracing the Republican candidate (on his right) and the Democratic candidate (on his left)

prefer Jewish narratives,
I recommend

 1. Kesher Talk’s “Dick Morris:
Flaming Sword of Vengeance

2. Eyes on the Prize

3. Triangulation.

Saturday, August 16, 2008

Saturday August 16, 2008

Filed under: General,Geometry — m759 @ 8:00 AM

Seeing the Finite Structure

The following supplies some context for remarks of Halmos on combinatorics.

From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–

Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–

“Part II: In Touch with God*“– on pp. 27-28:

The Root of All Deep Mathematics

Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.

Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.

Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.

Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”

Finite Structure
 on a Book Cover:

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
by F. Schipp et al.,
Taylor & Francis, 1990

Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.

Walsh Series
states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”

The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.

Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.

* See the Log24 entries following Halmos’s death.

Thursday, August 14, 2008

Thursday August 14, 2008

Filed under: General,Geometry — Tags: — m759 @ 4:19 AM
'Magister Ludi,' or 'The Glass Bead Game,' by Hermann Hesse
Magister Ludi
(The Glass Bead Game)
is now available for
download in pdf or
text format at Scribd.

“How far back the historian wishes to place the origins and antecedents of the Glass Bead Game is, ultimately, a matter of his personal choice. For like every great idea it has no real beginning; rather, it has always been, at least the idea of it. We find it foreshadowed, as a dim anticipation and hope, in a good many earlier ages. There are hints of it in Pythagoras, for example, and then among Hellenistic Gnostic circles in the late period of classical civilization. We find it equally among the ancient Chinese, then again at the several pinnacles of Arabic-Moorish culture; and the path of its prehistory leads on through Scholasticism and Humanism to the academies of mathematicians of the seventeenth and eighteenth centuries and on to the Romantic philosophies and the runes of Novalis’s hallucinatory visions. This same eternal idea, which for us has been embodied in the Glass Bead Game, has underlain every movement of Mind toward the ideal goal of a universitas litterarum, every Platonic academy, every league of an intellectual elite, every rapprochement between the exact and the more liberal disciplines, every effort toward reconciliation between science and art or science and religion. Men like Abelard, Leibniz, and Hegel unquestionably were familiar with the dream of capturing the universe of the intellect in concentric systems, and pairing the living beauty of thought and art with the magical expressiveness of the exact sciences. In that age in which music and mathematics almost simultaneously attained classical heights, approaches and cross-fertilizations between the two disciplines occurred frequently.”

 — Hermann Hesse

Author’s dedication:

to the Journeyers
to the East

Related material:

The Ring of the Diamond Theorem

Ring Theory

Monday, August 11, 2008

Monday August 11, 2008

Filed under: General,Geometry — m759 @ 9:00 PM
 New Illustration
for the four-color
decomposition theorem:

Four-color decompostion applied to the 8-point binary affine space

Sunday, August 10, 2008

Sunday August 10, 2008

Filed under: General,Geometry — m759 @ 10:31 AM

One Year Ago
in this journal —

Commentary by Richard Wilhelm
on I Ching Hexagram 32:

Hexagram 32, Duration, of the I Ching


“Duration is… not a state of rest, for mere standstill is regression. Duration is rather the self-contained and therefore self-renewing movement of an organized, firmly integrated whole [click on link for an example], taking place in accordance with immutable laws and beginning anew at every ending.”

Richard Wilhelm's grave. Note the eight I Ching trigrams.

Richard Wilhelm’s grave:
Note the eight I Ching
trigrams surrounding
the globe.

Globe at opening of 2008 Beijing Olympics

Globe at the
Beijing 2008 Olympics
Opening Ceremony

The eight trigrams
were perhaps implied in
the opening’s date, 8/8/8.

Friday, August 8, 2008

Friday August 8, 2008

Filed under: General,Geometry — Tags: — m759 @ 8:08 AM
Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Click on image for details.

Wednesday, August 6, 2008

Wednesday August 6, 2008

Filed under: General — m759 @ 12:00 PM

From the last link within the last link of yesterday’s entry:

Review the concepts of integritas, consonantia, and claritas in Aquinas….”

Review also the properties of the number six that appears in today’s date.

For such properties, see the page of Log24 entries that end on September 6, 2006, with “Hamlet’s Transformation.”

Happy Feast of the Transfiguration.

Tuesday, August 5, 2008

Tuesday August 5, 2008

Filed under: General — Tags: — m759 @ 2:02 PM
Published Today:

Cover of  'The Last Theorem,' a novel by Arthur C. Clarke and Frederik Pohl

The Last Theorem
, a novel by
Arthur C. Clarke and Frederik Pohl

From the publisher's description:

"The Last Theorem is a story of one man’s mathematical obsession, and a celebration of the human spirit and the scientific method. It is also a gripping intellectual thriller….

In 1637, the French mathematician Pierre de Fermat scrawled a note in the margin of a book about an enigmatic theorem: 'I have discovered a truly marvelous proof of this proposition which this margin is too narrow to contain.' He also neglected to record his proof elsewhere. Thus began a search for the Holy Grail of mathematics– a search that didn’t end until 1994, when Andrew Wiles published a 150-page proof. But the proof was burdensome, overlong, and utilized mathematical techniques undreamed of in Fermat’s time, and so it left many critics unsatisfied– including young Ranjit Subramanian, a Sri Lankan with a special gift for mathematics and a passion for the famous 'Last Theorem.'

When Ranjit writes a three-page proof of the theorem that relies exclusively on knowledge available to Fermat, his achievement is hailed as a work of genius, bringing him fame and fortune…."

For a similar third-world fantasy about another famous theorem, see the oeuvre of Ashay Dharwadker.

Note the amazing conclusion of Dharwadker's saga (thus far)–

Dharwadker devises a proof of the four-color theorem that leads to…

Grand Unification
of the Standard Model
with Quantum Gravity!

For further background, see

Ashay Dharwadker
  and Usenet Postings.

Clarke lived in Sri Lanka (formerly Ceylon) from 1956 until his death last March.

For another connection with Sri Lanka, see

Location, Location, Location
(July 13, 2005) and
Bagombo Snuff Box
(May 7, 2006).

Monday, August 4, 2008

Monday August 4, 2008

Filed under: General — m759 @ 9:57 AM

Summer of ’36

Another Opening
of Another Show

“When I cast my mind back to that summer of 1936 different kinds of memories offer themselves to me. We got our first wireless set that summer– well, a sort of a set; and it obsessed us. And because it arrived as August was about to begin, my Aunt Maggie– she was the joker of the family– she suggested we give it a name. She wanted to call it Lugh after the old Celtic God of the Harvest. Because in the old days August the First was La Lughnasa, the feast day of the pagan god, Lugh; and the days and weeks of harvesting that followed were called the Festival of Lughnasa.”

— Michael in the play
 “Dancing at Lughnasa”

From the film “Contact”–

Jodie Foster in 'Contact' viewing the opening of the 1936 Olympics

Jodie Foster and the
opening of the 1936 Olympics

“Heraclitus…. says: ‘The ruler
 whose prophecy occurs at Delphi
 oute legei oute kryptei,
 neither gathers nor hides,
 alla semainei, but gives hints.'”
 — An Introduction to Metaphysics,
 by Martin Heidegger, Yale University
 Press paperback, 1959, p. 170

Sunday, August 3, 2008

Sunday August 3, 2008

Filed under: General — m759 @ 10:00 PM
This Hard Prize

Triangle (percussion instrument)


"Credences of Summer," VII,

by Wallace Stevens, from
Transport to Summer (1947)

"Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed
The object, grips it
     in savage scrutiny,
Once to make captive,
     once to subjugate
Or yield to subjugation,
     once to proclaim
The meaning of the capture,
     this hard prize,
Fully made, fully apparent,
     fully found."


Lughnasa — An Irish harvest festival.

"It was usually celebrated on the nearest Sunday to August 1st." —Chalice Centre

Related material:

  1. Dancing at Lughnasa, a play by Brian Friel
  2. Natasha's Dance, an entry in this journal
  3. Dancing at Lughnasa, an entry in this journal from August 3, 2003
"Going up."
— Nanci Griffith   

Sunday August 3, 2008

Filed under: General — Tags: — m759 @ 7:20 PM
Note for a Triangle

Triangle (percussion instrument)
The triangle,
a percussion instrument
featured prominently in
the Tom Stoppard play
"Every Good Boy
Deserves Favour


From a BBC News webpage
last updated at 22:31 GMT
(6:31 PM EDT)
Sunday, 3 August 2008 —


Alexander Solzhenitsyn
dies at 89

http://www.log24.com/log/pix08A/080803-Solzhenitsyn.jpg"Russian writer Alexander Solzhenitsyn, who exposed Stalin's prison system in his novels and spent 20 years in exile, has died near Moscow at the age of 89.

The author of The Gulag Archipelago and One Day In The Life Of Ivan Denisovich, who returned to Russia in 1994, died of either a stroke or heart failure.

The Nobel laureate had suffered from high blood pressure in recent years.

After returning to Russia, Solzhenitsyn wrote several polemics on Russian history and identity.

His son Stepan was quoted by one Russian news agency as saying his father died of heart failure, while another agency quoted literary sources as saying he had suffered a stroke.

He died in his home in the Moscow area, where he had lived with his wife Natalya, at 2345 local time (1945 GMT) [3:45 PM EDT], Stepan told Itar-Tass.

Russian President Dmitry Medvedev sent his condolences to the writer's family, a Kremlin spokesperson said…."


Related material:

Today's 3 PM (EDT) entry.

Sunday August 3, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 3:00 PM

Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein’s birthday):

The play’s title, “Every Good Boy Deserves Favour,” is a mnemonic for the notes of the treble clef EGBDF.

The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.

The “u” in favour is the British way, the Stoppard way, “EGBDF” being “a Play for Actors and Orchestra” by Tom Stoppard (words) and André Previn (music).

And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.

When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard “to write something which had the need of a live full-time orchestra onstage,” the 36-year-old playwright jumped at the chance.

One hitch: Stoppard at the time knew “very little about ‘serious’ music… My qualifications for writing about an orchestra,” he says in his introduction to the 1978 Grove Press edition of “EGBDF,” “amounted to a spell as a triangle player in a kindergarten percussion band.”

Jerry Tallmer in The Villager, March 12-18, 2008

Review of the same play as presented at Chautauqua Institution on July 24, 2008:

“Stoppard’s modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience.”

Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008

“The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul.”

— Dan Fogelberg

“He’s watching us all the time.”

Lucia Joyce


Finnegans Wake,
Book II, Episode 2, pp. 296-297:

I’ll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you’d wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun!

1 The chape of Doña Speranza of the Nacion.


Log 24, Sept. 3, 2003:
From my entry of Sept. 1, 2003:

“…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, ‘Only connect.’ ‘Reciprocity’ would be Michael Kruger’s succinct philosophy, with all that the word implies.”

— William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994

Last year’s entry on this date:


Today’s birthday:
James Joseph Sylvester

Mathematics is the music of reason.”
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase “synthematic totals” to describe some structures based on 6-element sets that R. T. Curtis has called “rather unwieldy objects.” See Curtis’s abstract, Symmetric Generation of Finite Groups, John Baez’s essay, Some Thoughts on the Number 6, and my website, Diamond Theory.


The picture above is of the complete graph K6 …  Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester’s synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites….  “Reciprocity” in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of “reciprocity” more closely related to Michael Kruger’s alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger’s novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and


Salmonson set his seel:

“Finn MacCool ate the Salmon of Knowledge.”


George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest.”

Related material:

Kindergarten Theology,

Kindergarten Relativity,

Arrangements for
56 Triangles

For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.

Happy birthday,
Martin Sheen.

Saturday, August 2, 2008

Saturday August 2, 2008

Filed under: General — m759 @ 3:52 PM
Ready When
You Are, C. B.

(continued from
June 23, 2007)

Front page top center,
online New York Times,
3:12 PM Saturday, August 2, 2008:



Finlay MacKay for The New York Times

For gold-medal hopefuls like Nastia Liukin, there’s just one big chance to make it as a marketing darling.

Possible titles
for the above photo:

The Eye of Apollo
The Hidden Sign.

See also the conclusion
of the Wallace Stevens
poem linked to in
the previous entry.

Saturday August 2, 2008

Filed under: General,Geometry — m759 @ 2:02 PM
Geometry and Death

(continued from
June 15, 2007)

Today is the anniversary
of the 1955 death of poet
Wallace Stevens.

Related material:

A poem by Stevens,

an essay on  the
relationships between
poets and philosophers —
“Bad Blood,” by
Leonard Michaels


The ninefold square, a symbol of Apollo

the Log24 entries
of June 14-15, 2007

Saturday August 2, 2008

Filed under: General,Geometry — m759 @ 6:23 AM

There is an article in today’s Telegraph on mathematician Simon Phillips Norton– co-author, with John Horton Conway, of the rather famous paper “Monstrous Moonshine” (Bull. London Math. Soc. 11, 308–339, 1979).
“Simon studies one of the most complicated groups of all: the Monster. He is, still, the world expert on it ….

Simon tells me he has a quasi-religious faith in the Monster. One day, he says, … the Monster will expose the structure of the universe.

… although Simon says he is keen for me to write a book about him and his work on the Monster and his obsession with buses, he doesn’t like talking, has no sense of anecdotes or extended conversation, and can’t remember (or never paid any attention to) 90 per cent of the things I want him to tell me about in his past. It is not modesty. Simon is not modest or immodest: he just has no self-curiosity. To Simon, Simon is a collection of disparate facts and no interpretative glue. He is a man without adjectives. His speech is made up almost entirely of short bursts of grunts and nouns.

This is the main reason why we spent three weeks together …. I needed to find a way to make him prattle.”

Those in search of prattle and interpretive glue should consult Anthony Judge’s essay “Potential Psychosocial Significance of Monstrous Moonshine: An Exceptional Form of Symmetry as a Rosetta Stone for Cognitive Frameworks.”  This was cited here in Thursday’s entry “Symmetry in Review.”  (That entry is just a list of items related in part by synchronicity, in part by mathematical content. The list, while meaningful to me and perhaps a few others, is also lacking in prattle and interpretive glue.)

Those in search of knowledge, rather than glue and prattle, should consult Symmetry and the Monster, by Mark Ronan.  If they have a good undergraduate education in mathematics, Terry Gannon‘s survey paper “Monstrous Moonshine: The First Twenty-Five Years” (pdf) and book– Moonshine Beyond the Monster— may also be of interest.

Friday, August 1, 2008

Friday August 1, 2008

Filed under: General — m759 @ 2:42 PM
A Two-Part Invention
for Sarah Silverman

Part I:

(Thanks to
The Unapologetic Mathematician)

The puzzle



Part II:

The moves

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8/1/2008 11:30 AM
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(From this weblog’s footprints today)

Friday August 1, 2008

Filed under: General — m759 @ 2:56 AM
Front page, online New York Times,  Friday, August 1, 2008, 2:16 AM

Click on image for context.

Time of entry: 2:56:04 AM.

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